Semi-analytical Ansatz for Approximating Dynamics of Asteroid Rotation Approaching Planet

Document Type : Research Paper

Author
Sergey V. Ershkov 1 , 2
1 Sternberg Astronomical Institute, M.V. Lomonosov's Moscow State University, 13 Universitetskij prospect, Moscow 119992, Russia
2 MIREA - Russian Technological University, 78 Vernadsky Avenue, Moscow 119454, Russia
10.22055/jacm.2025.48493.5273
Abstract
The main motivation of this research is the analytical exploration of the dynamics of asteroid rotation approaching planet, in view of future approach Apophis 2029 to Earth. As previously noted, various perturbations (collisions, close encounters, YORP effect) may destabilize the rotation of minor celestial body (asteroid), deviating it from the current spin-state. We consider here semi-analytical algorithm for calculating regimes of asteroid rotation assuming its being not rubble-pile, but preferably of rigid state of its surface (this would assume almost constant main moments of inertia for asteroid, as first approximation). Main Euler equations stemming from assumption of conservation of angular momentum have been presented using applied gravitational torques during approach asteroid to planet. New method for solving Euler’s equations for rigid body rotation is introduced herein; an elegant example of evolution of non-linear spin dynamical state is demonstrated along with semi-analytical findings of kinematic presentation of sequent rotation stages via Euler (or Wisdom) angles.
Keywords
Tidal dissipation
Asteroid rotation
Gravitational torques
YORP effect
Apophis approach 2029
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Ershkov, S.V., Shamin, R.V., The dynamics of asteroid rotation, governed by YORP effect: the kinematic ansatz, Acta Astronautica, 149, 2018, 47–54.
[2] Ershkov, S.V., Leshchenko, D.D., On the dynamics OF NON-RIGID asteroid rotation, Acta Astronautica, 161, 2019, 40–43.
[3] Kartashov, E.M., Developing generalized model representations of thermal shock for local non-equilibrium heat transfer processes, Russian Technological Journal, 11(3), 2023, 70-85.
[4] Walsh, K.J., Rubble pile asteroids, Annual Review of Astronomy and Astrophysics, 56, 2018, 593-624.
[5] https://science.nasa.gov/science-news/science-at-nasa/2014/06mar_asteroid.
[6] Chapman, C.R., Morrison D., Zellner, B., Surface properties of asteroids: A synthesis of polarimetry, radiometry, and spectrophotometry, Icarus, 25, 1975, 104-130.
[7] Melnikov, A.V., Rotational Dynamics of Asteroids Approaching Planets, Solar System Research, 56, 2022, 241–251.
[8] Lobanova, K.S., Melnikov, A.V., Disturbances in the Rotational Dynamics of Asteroid (99942) Apophis at its Approach to the Earth in 2029, Solar System Research, 58, 2024, 208–219.
[9] Klavetter, J.J., Rotation of Hyperion. 2. Dynamics, Astronomical Journal, 98(5), 1989, 1855.
[10] Wisdom, J., Peale, S.J., Mignard, F., The chaotic rotation of Hyperion, Icarus, 58(2), 1984, 137-152.
[11] Pravec, P., Harris, A.W., Michalowski, Asteroid rotations. In: Bottke Jr., W.F., Cellino, A., Paolicchi, P., Binzel, R.P. (Eds.), Asteroids III, University of Arizona Press, Tucson, 113–122, 2002.
[12] Scheeres, D.J., Orbital Motion in Strongly Perturbed Environments. Applications to Asteroid, Comet and Planetary Satellite Orbiters, Springer, Praxis Publishing, Chichester, UK, 2012.
[13] Walsh, K.J., Richardson, D.C., Michel, P., Spin-up of rubble-pile asteroids: Disruption, satellite formation, and equilibrium shapes, Icarus, 220, 2012, 514–529.
[14] Melnikov, A.V., Shevchenko, I.I., Unusual rotation modes of minor planetary satellites, Solar System Research, 41, 2007, 483–491.
[15] Ershkov, S.V., Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation, Theoretical and Applied Mechanics Letters, 7(3), 2017, 175-178.
[16] Melnikov, A.V., Resonant and chaotic phenomena in the dynamics of celestial bodies. Spb, GAO Pulkovo Observatory, Dissertation thesis for the degree of the Doctor of Science, 2016.
[17] Szebehely, V., Theory of Orbits. The Restricted Problem of Three Bodies, Yale University, New Haven, Connecticut, Academic Press, New-York and London, 1967.
[18] Landau, L.D., Lifshitz, E.M., Course of Theoretical Physics, Mechanics, Volume 1 (3rd Edition), §37 ("The asymmetrical top"), Butterworth- Heinemann, Linacre House, Jordan Hill, Oxford, 1976.
[19] Vokrouhlicky, D., Bottke, W.F., Chesley, S.R., Scheeres, D.J., Statler T.S., The Yarkovsky and YORP Effects, at http://arxiv.org/abs/1502.01249, 2015.
[20] Chernous’ko F.L., On the Motion of a Satellite about Its Center of Mass under the Action of Gravitational Torques, Prikl. Mat. Mekh., 27 (3), 1963, 474-483 [J.  Appl. Math. Mech. (Engl. Transl.) 27(3), 1963, 708–722].
[21] Chernousko, F.L., Akulenko, L.D., Leshchenko, D.D., Evolution of motions of a rigid body about its center of mass, Springer, Cham, 2017.
[22] Beletsky, V.V., Motion of an Artificial Satellite about its Center of Mass, Israel Program for Scientific Translation, Jerusalem, 1966.
[23] https://www.integral-calculator.com/
[24] Benson, C.J., Scheeres, D.J., Brozovic, M., Chesley, S. R., Pravec, P., Scheirich, P., Spin state evolution of (99942) Apophis during its 2029 Earth encounter, Icarus, 390, 2023, 115324.
[25] Souchay, J., Lhotka, C., Heron, G., Herve, Y., Puente, V., Lopez, M.F., Changes of spin axis and rate of the asteroid (99942) Apophis during the 2029 close encounter with Earth: A constrained model, Astronomy & Astrophysics, 617, 2018, A74.
[26] Holsapple, K.A., Michel, P., Tidal disruptions: A continuum theory for solid bodies, Icarus, 183(2), 2006, 331-348.
[27] Hirabayashi, M., Kim, Y., Brozovic, M., Finite element modeling to characterize the stress evolution in asteroid (99942) Apophis during the     2029 Earth encounter, Icarus, 365, 2021, 114493.
[28] Leshchenko, D., Ershkov, S., Kozachenko, T., Evolution of a heavy rigid body rotation under the action of unsteady restoring and perturbation torques, Nonlinear Dynamics, 103, 2021, 1517–1528.
[29] Ershkov, S.V., Shamin, R.V., On metallic-type asteroid rotation moving in magnetic field (introducing magnetic second-grade YORP effect), Acta Astronautica, 224, 2024, 195–201.
[30] Ershkov, S.V., Revisiting dynamical orbits in the planar anisotropic Kepler problem, International Journal of Non-Linear Mechanics, 173, 2025, 105029.
Journal of Applied and Computational Mechanics
Volume 12, Issue 2
Spring 2026
Pages 483-490